QA-P2: Euler-Poncelet Point
Euler mentioned this point in one of his numerous papers.
In 1821 Brianchon and Poncelet, both captains of artillery, wrote a book (Ref-1) in which the orthogonal hyperbola and this point were worked out.
The Euler-Poncelet Point can be defined in different ways.
1. It is the center of the orthogonal hyperbola through P1, P2, P3 and P4.
2. It is the common point of the Nine-point Circles of the triangles Pj.Pk.Pl for all permutations of (j,k,l) ∈ (1,2,3,4).
3. It is the common point of the Pedal Circles of points Pi wrt triangles Pj.Pk.Pl for all permutations of (i,j,k,l) ∈ (1,2,3,4).
p (SB q - SC r) (b2 r (p+q) - c2 q (p+r))
1/(b2 r2 - c2 q2)
- QA-P2 lies on these QA-lines:
- QA-P2 is the center of the Orthogonal Hyperbola through P1, P2, P3, P4.
- QA-P2 is the common point of Nine-point Circles Pi.Pj.Pk for all permutations of (j,k,l)∈ (1,2,3,4).
- QA-P2 is the Homothetic Center of the Antigonal Quadrangle and the Reference Quadrangle (the Antigonal of a point X isthe isogonal conjugate of the inverse in the circumcircle of the isogonal conjugate of X, see Ref-17a.
- QA-P2 is the Midpoint of the reflections of QA-P4 in Pi.Pj and Pk.Pl (note Eckart Schmidt).
- Let M = Diagonal Point Pi.Pj ^ Pk.Pl. Now M.QA-P4 and M.QA-P2 are symmetric wrt the angle bisector of lines Pi.Pj and Pk.Pl for all permutations of (i,j,k,l) ∈ (1,2,3,4). See Ref-16 page 8.
- QA-P2 lies on the circumcircle of the QA-Diagonal Triangle.
- QA-P2 lies on the circumcircles of triangles formed by the 3 QA-versions of QG-P1, QG-P6, QG-P10 as well as QG-P14.
- QA-P2 lies on the Nine-point Conic (QA-Co1).
- QA-P2 is concyclic with QA-P7, QA-P8 and QA-P23.
- A Maltitude ("Midpoint altitude") is a perpendicular drawn to a side of a Quadrigon from the midpoint of the opposite side. The 2nd generation of the Maltitude Quadrigon is homothetic with the Reference Quadrigon. Let M be their perspector. M is the same point for each quadrigon in a Quadrangle and coincides with QA-P2.
QA-P2 is also the Euler-Poncelet Point of the Orthocenter Quadrangle (Eckart Schmidt, August 24, 2012).
- QA-P2 is the Orthopole (see Ref-13) of line QA-P4.Pi wrt triangle Pj.Pk.Pl for all (i,j,k,l) ∈ (1,2,3,4). See Ref-11, Hyacinthos messages # 21865 & 21867.
Let O1, O2, O3, O4 be the circumcenters of the Component Triangles of Reference Quadrangle P1.P2.P3.P4. Let OPi be the Orthopole of line P.Oi (P=random point) wrt Component Triangle Pj.Pk.Pl, where (i,j,k,l) ∈ (1,2,3,4). The 4 Orthopoles OP1, OP2, OP3, OP4 are concyclic on a circle through QA-P2. See Ref-11, Hyacinthos messages # 21865 & 21867.